Estimating and Number Lines

This week we were focused on two things:  estimating stuff and counting to see if our estimate was close.  I’m feeling really good about it!

There were some fun activities we did that I think really helped.

First, I had some small jars full of stuff.  We started the week by reading a book about estimation.  Then I held up one jar at a time and asked everyone to estimate how many were in the jar.  After the second one, I sent them to their tables to practice.  They had a great time estimating how many paper clips, beads, erasers or rocks were in each jar.  We did this on Tuesday and on Wednesday.  (We didn’t have school on Monday.)

After we had counted them, one of my friends suggested that maybe I had 47.  Win some, lose some, right?  But I put that on the number line and we discussed our answer of 37 again and I think that friend understood that I had 37 and how far away from our estimate 47 is.

An interesting thing happened while we were counting.  Thirty-seven is a high number for some kids to hold in their head so they were using fingers and counting out loud to aide their working memory. I wanted to talk about this strategy so that those who hadn’t done it would know it’s a strategy they could use.  One friend said that he had actually only been able to count to 10 on his fingers at first so each time I got to ” a group of 10″ (“Like 10, 20, 30…like that!”) counters he held up 1 finger. He knew he had 3 fingers and that is 30 counters, then he just had the 7 to go with the 30.  I tried to draw that thinking too.  This strategy actually lead really nicely into our lesson.  We are working on the “Collecting and Organizing” Context for Learning unit next, and counting stuff is the beginning of that unit.  He introduced to us the idea that things can be put into groups of 10 to help with organizing and counting.

We did a bit more counting on Friday.  Everyone tried to make groups of ten, but many aren’t yet convinced that this will help.  We’ll dive deep into this unit whenever we go back to school (hopefully Monday!) and I feel confident they will have it by the end.

We finished on Friday with the “Flying Cars” Esti-Mystery from Steve Wyborney’s new Esti-Mystery set.  It was a huge success and the students were so excited that their estimate was so close to the real answer.  I was so excited that their ability to both reason and explain their reasoning had come so far in just one week.

Up next on the spiralling document I have been following is more counting (forward to 100 for grade 2 and 200 for grade 3).  This week we did some hundred chart puzzles.  I had some made with 101-200 charts for the grade 3s to work on.  They all did pretty well.  They can now become a centre when I need everyone to do independent activities while I run Guided Math groups.  This will become really important in about 2 weeks (depending on if/how long schools are closed for the strike) when I want my grade 3s and grade 2s working on some different units. We also need to be able to count backward (from 50 for grade 2 & 3, and from 500 by 100s for grade 3s) so that will be the focus of our counting routines next week.

And look….nobody went to the washroom during our Number Talk that day!  Interpret that as you will.

I can’t sum up this week in a title

Classes reorganized this past week.  I had a new class list on Monday, and about 1/2 of  my class is new since the first day of school. The dynamic shifted.  Though it wasn’t exactly like starting over, it was enough like starting over that I didn’t feel like I could move forward as quickly as I wanted to.  I decided, therefore, to do activities that would sort of take us back in time to the first week while also moving us forward.

These are three of my first 2 weeks goals that I needed to accomplish in just this past week – the 3rd week:  really work on how to use the manipulatives without driving our teacher bonkers, work on what to do during Number Talk/Number Sting time, and practice working respectfully with a partner.  The friends who stayed with me had already practiced this for 2 weeks so I couldn’t simply start over.  And they didn’t need me to anyway. The friends who moved in had already practiced some of this with their “first 2 weeks” teacher, but we needed to do it the Room 16 way.  I realize that makes it sound like I have some control issues, but I assure you I do not.  Really.  I don’t. Please believe me!

So…here’s what we did.

Find it on the 100 chart: I gave them hundred charts, in groups of two.  I gave them containers full of counters.  I called out clues, from “Find it on the 100 chart”  by Marcie Cook which I have owned for approximately 100 years.  I called out clues, they had to cover the number I clued and in the end the manipulatives would have built a picture.  They had to cooperate with their partner and share turns.  They had to clean everything up, even off the floor, when we finished, and put it all away.

Counting Necklaces:  You’d think they’d be getting tired of this by now.  But they are not.  Not even close.  I’ve written about it before here, so I’m just going to say that we ended our math class with this all week and by the end of the week nobody was getting too upset about not getting a necklace because they realized there would be more turns on another day. Once again we had many opportunities to talk about what patterns we were seeing, and what happens to the pattern if someone leaves the circle to use the washroom or if someone moves to another spot.

Usually by this time on a Saturday morning I have a goal for the following week. In fact, I always start writing my lesson plans by writing my math plans for the week.  I’m not ready today.  We accomplished some good goals for the week, but we needed more repetition to get there.  That means I am not where I wanted to be on my curriculum map.  I’m generally okay with that, but I also think there might be a way to still get there by next Friday as planned.  I’m following the TIPS spiralled math document and even though I know I need to be flexible with the timing, I also want to try to trust the timing.  We’ll come back to all of our content again. I don’t need to teach to mastery the first time for every skill.  I think I’ll move forward with our 100/200 chart puzzles.  I think we’ll move forward with the estimation and counting jars/bags.  I think it’s a 4 day week with no other interruptions so we should be able to do that.  Maybe I am more ready to write my plan than I thought.

Summer Math: Math Before Bed

We love doing “Math Before Bed” as part of our “read at bedtime” routine.  We get out of the habit sometimes though because we also love to play card games (UNO, Go Fish, Old Maid, Memory) before bed. Last night I pulled up this picture:

I quickly counted them:  10 per column, 4 in each row. 40.

Then she asked, “Can I make my own picture like this tomorrow?”  So that is what we have just finished.  She decided to use plasticine.  I was recruited to mix colours together and help her make tiny balls.  She decided she needed 60 of them. She also decided she wanted to do rows of three because 2’s and 5’s are too easy and she likes a challenge. (HOORAY!!!)  After counting over and over by 3’s, making a few mistakes along the way, I prompted her to notice that there were 10 in a column.  “10, 20, 30.  Oh.  Halfway there.”  🙂

In the end, we had more than we needed. She put those into groups of 5 (and one group of 4) to figure out how many were left. “5, 10, 14,” she said.  It’s so interesting to me that she can skip count, but often counts by ones.  She says this is because “ones is more easier.” She only switches to larger numbers and skip counting when she has a lot of things to count. I suppose this makes sense.

Summer Math:Counting and Subitizing

I’ve been in a “blogging about math” funk for a few months.  One of my summer goals is to write more, so I thought I’d start a series of blog posts about the math that I am doing with my children at home this summer.  To be clear, this is not a series that is planned.  Instead, I am going to try to be very mindful of the times we do math together formally or informally.  My children, who just finished grade 1 and 2, are probably involved in informal math conversations about the same as many children of teachers. Both of them are pretty good mathematicians, and by that I mean they use flexible strategies to do mental math calculations, they notice math in the world around them, and come up with strategies for solving math problems that naturally occur around them.  I’m a firm believer that this happened because I am intentional about helping them mathematize their world, just as I am intentional about making sure they learned to read by reading to them at least every night before bed (and usually more often!)

Earlier in the week I received an e-mail about the latest Mathies resources.  This morning we finally had time to sit down and explore a bit.  I picked a game for my 6 year old called “Representation Match” and if it wasn’t for his Minecraft addiction I think we’d still be playing it (he only gets to play on Saturdays and when he sneaks my phone into his closet unnoticed so it’s tough competition!)

I chose the numbers 0-20 for him, and I chose all the representations of those numbers.  He had to find matches – two ways to make 14, or 19, or 17, or any number between 0-20. These are all the choices available.

He had to work at this!  He was not able to subitize all the numbers so we had a few conversations about how to figure out the number represented.  For example,  there were 3 dice, two showing 6 and one showing 5. I prompted him to think about 6+6 which he knows is 12.  Then I pointed to the 5. He counted on by 1’s to get to 17, then chose the numeral 17 as it’s match. Sometimes he had to match two picture representations.

When he played again, he chose 2 or 3 representations for himself, always a different combination.  My daughter did the same.  She played with the 0-20 cards, even though she is in grade 2.  She likes to get answers fast, so this appealed to her. She was also playing with the cards hidden, more like a traditional memory game and said she had a lot to keep track of in her mind if she was playing with the higher numbers.

We use Dreambox a lot at school.  I love it!  But I also like to have students doing some targeted math activities that keep them immersed in a specific skill for a while.  Dream box allows them to pause a game and move on to something else, which is fine, but also lets them give up too easily sometimes.  I think this Mathies game would make a great supplemental activity for us during the first month or more of school when we are talking about counting strategies, as well as for practice throughout the year.  Did I tell you I’m scheduled to teach a grade 1, 2, 3 split next year? I haven’t taught grade 1 before so I am anticipating how that will look.  The “Representation Match” game will let me set them up to match numbers 0-5, 0-10, 0-20 and 20-50, I think it will be good for the whole class.

This past week at OAME, I was pretty focused on spiralling the math curriculum and on finding more problem solving tasks to use with my class.  I find that a lot of the tasks are a bit beyond our reach, which is frustrating.

One of the things I was introduced to was Graham Fletchy’s 3 Act Math Tasks. I so appreciate when a person is willing to create a resource like this and then share it with the wide world!  While planning my week, I picked out a few in particular that I thought would engage my students, while also spiralling us back to some Big Ideas we haven’t worked with for a while.

Today we did this task, called Snack Machine.  We have had a lot of practice working with each other.  We have had a lot of practice thinking about a strategy to use to solve a problem.  But this task, and others on the site, really allow for a lot of divergent thinking.   There are multiple entry points, and multiple paths to a solution.  It’s great!

In the Snack Machine, a video shows a girl buying something from a vending machine.  We watched, then talked about it, then watched again, then talked again.

At this point, the children didn’t know what the problem would be.  They were simply looking at the video and mathematizing it. The discussion started off with someone suggesting that the girl in the video looked at her change and was disappointed.  That definitely had people thinking about why.  I got a kick (as my grandma would say) out of one of them suggesting that the machine scammed her.

After the second viewing, we had things to add.  We heard 4 coins fall, so which coins might they have been?  That lead to a long conversation, mainly because 4 toonies would make that sound, but would be an awful lot of money for a bag of chips, but 4 nickels wouldn’t really make sense either.  In act 2, there is a picture of the vending machine showing us that the chips actually cost 60 cents. Then another video shows the machine counting up the money.  We added that to our board:

After this, I sent them off to figure out the coins she must have used.  Amazing things happened!  After everyone had a pretty good shot at solving the problem, I showed the final video.  In that video we see that the change was 2 dimes.  They used this to confirm that 80 cents had gone in, 20 cents had come out + 60 cents worth of chips, so it all made sense. No scam!

The money used was American money, and of course a little bag of chips would cost more than 60 cents in a Canadian vending machine. But I told them the two coins we saw were dimes, and that was good enough for them.

Yesterday we worked on Sliced Up, which had us estimating, thinking backward from oranges cut into wedges to whole oranges, and finally multiplying (5 whole oranges, 4 wedges from each orange so how many wedges in all?) For tomorrow, I am debating between It All Adds Up which is a nice money connection to Snack Machine, and The Whopper Jar   which is a nice follow up to the estimating we did in Sliced Up.  Whichever problem doesn’t make the cut tomorrow will our Monday task.  I’m learning toward the money problem because I have a bunch of activities we could do as Number Talks to stretch that learning all week.

It’s EQAO week at our school and I like having some fun, confidence building task for my students to work on.

Slice of Life: Multiply the Money

I wrote yesterday about a Number Talk I had worked on with my class.

Today I extended that activity by making an array using money.  I used the Mathies money tool to display an array made of nickels, then I asked, “What do you see?”

We built this display of our thinking a bit at a time, so I am sure it made more sense to us than it might to you!  Someone saw 45 cents.  Then someone else saw 15 cents (3x5cents) and then 3 groups of 15 = 45.  Some saw the array 3 x 3 = 9 nickels all together. I pointed out 3 x 15 and 9 x 5.  Thanks to our work with fact families, a few realized 45 “shared by” 3 people means they get 15 cents each.  Seriously!  They didn’t just point out that 3×15=45 so 45 divided by 3 = 15.  They actually explained what was happening! (Full disclosure: not all of them.)

After this conversation, I cleared the board and made an array using toonies. Now, you might have been expecting me to use dimes, but I thought the \$2 coin was better considering that we haven’t talked about multiplying by 10 and my grade 2s would get a lot more out of the conversation if we were thinking about multiples of 2 rather than 10. I will probably give 10s a try tomorrow on their own.  After our success today it seems like a good way to get everyone to start thinking about multiples of 10. I’m looking forward to it!

What you see isn’t what I see

For Number Talks this past week I made some arrays on Smart Notebook and projected them on the board. We spent time each day talking about what we noticed and thought. Early on the fact families emerged. I’m glad because we’ve just finished up some multiplication and division learning and I was glad to see this being put into practice.

On Friday I displayed the picture below:

As you can see there were multiple ways to see this picture. Immediately people saw the array of 4 rows with 3 bikes in each.

One child kept insisting it was 2 groups of 6. It took a minute for him to convince his peers, and I had to help by circling the 2 groups. I’m glad we took the time to let him explain! He was clearly showing some beginning “partial products” thinking and I wouldn’t have known this if I hadn’t probed for an explanation.

Finally someone started talking about the bike tires. I don’t have a photo of that annotation, but it was interesting to see how the students went in to figure out that 12 groups of 2 = 12+12 = 24. Of course they were able to complete the fact family. At this point very few were counting by one’s. I was quite happy about that for sure!

It took me very little time to use the tools in Smart Notebook to make these arrays. I’ve definitely used Mathies for this as well, but the pictures in Smart Notebook led to a deeper conversation. This week I need to find some cars with 4 wheels to expand on our conversation.

This work also builds on the “Eyes on Math” number talks and picture-based number talks we’ve been doing all year. Tomorrow we are going on an array hunt around the school with our iPads. That activity has been preempted a few times but I think it will end up being a better activity now that we’ve discussed the picture arrays a few times.

Math at Home

My son, who is in grade 1, has really good number sense.  He has a lot of mental math strategies that he uses efficiently and flexibly.  He adds on, he counts back, he finds landmark numbers, he even splits numbers!  And no, this is not because we spend a bunch of time every day drilling math.  It’s because we play lots of games and have math conversations that pop up throughout our day.

As I watched him play “Sorry” I was surprised that he was having some counting trouble.  He has been able to count in sequential order with one-to-one tagging for quite some time. He can count a variety of object by ones, more than 100, and when he makes a mistake he notices it on his own and fixes it.  He subitizes, and I feel like this what he is doing  while he counts and that his how he notices his own mistakes.  But that’s a tangent I won’t go on right now.

What surprised me as we were playing “Sorry” this week was the trouble he was having  moving his pawn the correct number of spaces on the board.  He recognizes every number in this game, and connects the number symbol with the amount. He’s done this with other games many times, such as when we play other games and he has to compare which of two numbers is larger. (I had a hard time writing that sentence because I kept thinking about how we haven’t played War in a long time!)  When he drew 5, for example, I know he knows that is 1, 2, 3, 4, 5.

When he would draw a number he would count to that number as he bounced his pawn around the board, but invariably any time he had a number higher than 3 he would bounce a different number of spaces.  Sometimes he would go fewer than he was allowed, and sometimes he would go farther than he was allowed.  If you draw a 4 in this game, you have to go backward, and he did OK with that but he would count slower than usual, so I built that into my intervention. I told him about the problem.  “Just like when you are counting things, your pawn has to touch each square when you count it.” I started by putting his hand in mine, and making sure that every bounce had his pawn landing in just one box without skipping any boxes.  After several rounds of this, he started doing it on his own.  He would slow down his counting and he’d land in the right spot.

The next day we played again, and the problem resurfaced.  This time I explained the problem to  him, then instead of holding his hand I put a finger on the square as he counted.  If he got ahead of me, or skipped a square, he would recognize this on his own and correct himself (and sometimes his big sister had to butt in and point out his mistake, but that’s a different post altogether!)

The third time we played the game, he needed a verbal reminder, but that was it.  And the fourth time he needed the verbal reminder.  And if we have time to play it again tomorrow, which I hope we will, I expect he’ll need the reminder again, but I’ll wait and see.

This whole thing has surprised me some, mainly because as I said before he knows how to count with one-to-one tagging and has for a while.  So why was he having trouble? This is what I think: there was a little pressure on him this time that isn’t normally there. First, he loves to win and he knew that winning in this game requires getting around the board quickly.  That was a distraction and a stressor when he was trying to count. Second, besides just counting, there was some other thinking that had to happen.  If you land on a square with a triangle you get to slide, and if you land on a square that already has a pawn on it then you say “Sorry!” and bump that pawn back to start, and sometimes I could see that he was making a move with one pawn while also thinking about how maybe he should actually be moving a different pawn to get a better outcome. He’d be in the middle of a move, suddenly stop, put the pawn back where it was and move a different one instead.  Third, …I don’t actually have a third.  I think those two things are enough to explain why he was having some trouble. I did double check to make sure he was wearing his glasses the first time I noticed it, and he was, so we can’t blame the vision.  And his coordination is such that moving a pawn around the board is not a physical difficulty for him.

Counting is such an interesting thing, isn’t it? I feel like I have some new insight into him as a mathematician.  I have since noticed that he also needs reminders to slow down when he is doing calculations.  He also does a better job when it is just me and him and he doesn’t have to worry about his sister butting in with answers. (Are you noticing a theme here?  It’s hard to be the little brother!) Finally, he does a much better job and enjoys the whole thing more when he can do single step problems. I feel like that last part is developmental and will work itself out over time.

My diagnosis is that there is an executive functioning thing going on.  He is using his working memory to do multiple tasks each time he takes a turn, not the least of which is to manage his emotions around the fact that his big sister is always butting in.

I am, of course, thinking about how to help my son with this particular thing.  But what does this look like in a classroom?  I’m thinking it would be useful to sit down with a few of my students and play a round of “Sorry” or “Trouble” or even “Snakes and Ladders” and really play with them.  They do these sort of things sometimes during indoor recess, but if I were to set this as an activity during class it would be so a group of children would be busy while I work on the real math with other kids.

Time to rethink that practice.

Counting

Years ago I bought this treasure at a yard sale for \$1:

There are well over 100 beaded necklaces in that bin!  I use them exclusively for math, though I definitely have had some children in the past 10 years who would have loved to wear them, or just run their hands through them over and over.  (It does feel nice!)

I bought them to use for a specific counting game.  I didn’t know about this game until I came to Canada.  Seemed every Core French teacher I ever worked with loved this game, though now that I am in an Immersion/English dual track school it isn’t as popular.  In French, this game is called “Dix” or Ten. The class sits in a circle and counts to 10, each saying one number.  Whoever says ten gets to sit down, and the game is played until there is just one person left.  I bought these necklaces when I was teaching kindergarten.  I didn’t want anyone to get out because the “out” people aren’t getting any practice.  I feel like I may have read about this in the Effective Guide to Instruction in Mathematics, but I can’t be sure.

Over the years, this game has evolved. I now use it for skip counting by all sorts of numbers: count by 10s and whoever says 100 gets a necklace, count by 5s and whoever says 50 gets a necklace, and so on.  I am getting ready to start some multiplication with my class after the March Break, so last week I pulled out the necklaces and we started using them every day for a few minutes before the mini-lesson.

On Friday, I asked everyone to count by 10s, and whoever said 30 got a necklace.  After we’d made it around the circle once, I asked them to talk about the pattern they could see.  Several realized there was a pattern.  It was identified as a “no, no, yes” pattern an “ABBABB” pattern, and a “skip, skip, yes” pattern.  Finally someone said, “It goes, 1, 2, 3! 1, 2, 3!” (emphasis on the 3!) I asked what would happen if we counted by ones.  Sure enough, every time someone said 3 s/he was wearing a necklace.  Then we counted past 3 to see if the pattern would continue.  I scribed on the board for them so everyone could see the numbers while we counted, and then I circled the numbers that corresponded with a person wearing a necklace.

Sure enough!  The pattern continued.

We talked about how we could use what we had learned to count by threes, just like when we count by 5s or 10s or 2s.  Everyone was amazed, and several were happy to share their strategy: say the numbers you are skipping quietly to yourself then say the third number loud and proud.

I’ve been reading the book “Number Routines” by Jessica Shumway, and this activity shows up in that book too.  She recommends that the class start with one of her many number routines, then Number Talk, and then the mini lesson.  I’ve been giving that a try this week and I like the way the counting routine lead into the lesson, which is going to lead into our next unit of study.

Well, not exactly “next”.  We’re going to spend a bit of time on time and temperature.  But then it’s off to multiplication we go!

Early Algebra

We started the “Trades, Jumps and Stops” Context for Learning unit quite some time ago. A variety of inclement weather days has interrupted us a lot, so we are behind where I thought we’d be at this point. That might be part of the issue I’m having with this unit.  I’ve not taught it before, so that has to take a bit of the blame as well.  And finally, I’m thinking I might have misjudged our general readiness for this unit.  But I talked to my down-the-hall neighbour who is also doing the unit and she concurred:  It seems that some kids are easily getting it, and some are really, really having to work hard to get it.  There’s not a lot of in-between here.

Day 4 of the unit begins with a mini-lesson that we needed quite a bit of time with.  I was to fill 2 separate bags with a certain amount of coins in each, plus a “mystery” coin.  We have had a bit of trouble adding up coins, so I decided to stretch this out a bit instead of trying to get through it in 15 minutes.  Instead of putting the bags out of reach, I gave the coins to kids.  I started by explaining, “X and Y have some coins.  They have an equal amount of money, but they have different coins.  I want to see if you can figure out which coins they have.” On the first day only one child had a “mystery” coin (a poker chip!), but on the second day both kids had one.  The children holding the coins were very excited to be given this job. It gave them practice identifying the coin by name and value.  Have you ever thought about how we interchangeable use “dime” & “10 cents” when talking about coins? Some of the kids are still calling this “The Boat”.

We unpacked the bags a bit at a time.  I don’t have pictures of the whole process, but this is how it looked in the end.  Obviously our mystery coin was worth 10 cents this time.  This is the beginning of the children learning about a variable, and I think they did OK!

On the second day, we talked more about the signs < > and =.  We added up the coins in chunks, as suggested in the lesson, and then decided if we needed to have a < or a > or if we finally needed the =.  This time, instead of adding them up as a group, I had the students work with partners.  We knew that X had 40 cents, and Y had 5o cents, but how much would they have once they each got another quarter?  This partnership was trying to make a number line across the bottom end of this photo, without actually making the number line.  It’s a step in the right direction for them!

We had lots of people able to do this:

Finally we made it to the mystery coin.  We knew how much money everyone had in actual coins, but what was that mystery coin worth? This led to one of those really cool moments when I felt like (most) everyone was excited about the math.  You can see here where I recorded some of the different responses to the value of the mystery coin.

I was even able to use a double number line to show two of the answers, and since that is the goal of this unit (developing an understanding of the double number line is in fact the very next lesson we will do!) I felt very good about that.

The next day I needed a Number Talk to reinforce the understanding of the variables.  I didn’t really NEED to do it, since this doesn’t come into the curriculum until a later grade, but they were excited about it, and it certainly can’t hurt them so we did it anyway. I found some images on Math Before Bed and use them for our Number Talk.  I feel like they really help to reinforce the student’s number sense because there is more than one way to make 10, or 12, or any number really.

The weather here is terrible today (Sunday) but I’m hopeful there will be plenty of actual school days this week when we can move forward with this unit.  I don’t want to lose our momentum!  Next time I do this unit, however, I am going to maybe wait a bit.  Of course, the thing that keeps getting me through is that I have to trust Cathy Fosnot! She says this will work, and she has seen it work with many, many children, so I am going to go ahead and finish the unit.  The students are mostly getting the ideas behind the math. Some of them are actually in need of more practice with adding up money.  I am going to make sure we have a day this week when we have some activities that require counting money and we’ll rotate through those to give everyone lots of practice. Often these blogs help me think through what has happened and what needs to happen next!  Time to stop writing and plan my money counting activities.